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Consider the equation $$\beta x\text{erf}(x)=\frac{1}{\sqrt{\pi}}\exp(-x^2),$$ where $\beta\ll1$ and $$\text{erf}(x)=\frac{2}{\sqrt{\pi}}\int_0^xe^{-t^2}\text{d}t$$ is the error function. I am looking for an approximate solution of the form $$x\approx x_1+\delta x_2,$$ where $\delta=\delta(\beta)$. For instance, for $\beta\gg1$, one can find the solutions $x\approx\pm(2\beta)^{-1/2}$, but I don't know where to start with the case where $\beta$ is very small.

Thanks in advance!

Marc
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1 Answers1

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For small $\beta$, you can use the Taylor expansions of both sides. Specifically, $$ \beta x \cdot erf(x) - \frac{1}{\sqrt{\pi}}\exp(-x^2) = -\frac{1}{\sqrt{\pi }} + \frac{(2 \beta +1) x^2}{\sqrt{\pi }} + \frac{(-4 \beta -3) x^4}{6 \sqrt{\pi }} + O(x^6) $$ Setting this equal to 0 gives $$ x = \pm \frac{1}{\sqrt{1 + 2 \beta}} $$ to leading order.

Hans Engler
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